# Span of a state price

Good morning, I report here an equation that you can find in the following paper: "Portfolio Selection with Options and Transaction Costs" by Semyon Malamud (2014) page 10 - 11 (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2397760)

Suppose that the state price $$\eta_{\left(i,\theta\right)}$$ for a stock $$i$$ depends on both its own price and its observable volatility $$v_{i.t}$$ so that $$\eta_{\left(i,\theta\right)}\left(X_t,y\right)=\frac{1}{\sigma_t}\left(\left(y - X_{i,t}\right)\right)/v_{i,t}$$. Let $$k_{X_i}$$ and $$k_{v_i}$$ denote the components of $$k$$ corresponding to the derivatives with respect to $$X_i$$ and $$v_i$$. For any $$k$$ with only $$k_{X_i}$$ and $$k_{v_i}$$ non zero, I have:

$$$$\eta_{\left(i,\theta\right)}^{\left(\textbf{k}\right)}\left(X_t,y\right)=\frac{\partial^{k_{X_i}+k_{v_i}}}{\partial X_i^{k_{X_i}}\partial v_i^{k_{v_i}}}\frac{1}{\sigma_t}\eta_{\left(i,\theta\right)}\left(\frac{\left(y - X_{i,t}\right)}{v_{i,t}}\right) =\sum_{k\leq |\textbf{k}|}^{} B_k\left(X_t\right)\eta_{\left(i,\theta\right)}^{\left(\textbf{k}\right)}\left(\frac{\left(y - \bar{X}_{i,t}\right)}{v_{i,t}}\right)$$$$

Consequently the span of $$\eta_{\left(i,\theta\right)}^{\left(\textbf{k}\right)}\left(X_t,y\right)$$, $$k_{X_i}+k_{v_i}\leq k$$ has dimension $$k$$. If claims on volatility $$v_{i,t}$$ are tradable and the corresponding conditional state prices are given by:

$$$$\eta_{\left(v_i,\theta\right)}\left(X_t,y\right)=\eta_{\left(v_i,\theta\right)}\left(X_{i,t},v_{i,t},y\right)$$$$ the span of the corresponding derivatives will be larger.

The equation and the subsequent comments are quite obscure to me. The coefficients $$B_k\left(X_t\right)$$ are not even defined throughout the paper. I can just think of some application of a Taylor expansion.